Commutative semigroups with cancellation law: a representation theorem

Abstract

Any commutative, cancellative semigroup S with 0 equipped with a uniformity can be embedded in a topological group $\widetilde{S}$. We introduce the notion of semigroup symmetry T which enables us to turn $\widetilde{S}$ into an involutive group. In Theorem 2.8 we prove that if S is 2-torsion-free and T is 2-divisible then the decomposition of elements of $\widetilde{S}$ into a sum of elements of the symmetric subgroup $\widetilde{S}_{s}$ and the asymmetric subgroup $\widetilde{S}_{a}$ is polar. In Theorem 3.7 we give conditions under which a topological group $\widetilde{S}$ is a topological direct sum of its symmetric subgroup $\widetilde{S}_{s}$ and its asymmetric subgroup $\widetilde{S}_{a}$. Theorem 2.8 and Theorem 3.7 are designed to be useful tools in studying Minkowski–Rådström–Hörmander spaces (and related topological groups $\widetilde{S}$), which are natural extensions of semigroups of bounded closed convex subsets of real Hausdorff topological vector spaces.